Shard polytopes

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Shard polytopes

Arnau Padrol, Vincent Pilaud, Julian Ritter

To cite this version:

ARNAU PADROL, VINCENT PILAUD, AND JULIAN RITTER

Abstract. For any lattice congruence of the weak order on permutations, N. Reading proved that glueing together the cones of the braid fan that belong to the same congruence class defines a complete fan, called quotient fan, and V. Pilaud and F. Santos showed that it is the normal fan of a polytope, called quotientope. In this paper, we provide an alternative simpler approach to realize this quotient fan based on Minkowski sums of elementary polytopes, called shard polytopes, which have remarkable combinatorial and geometric properties. In contrast to the original construction of quotientopes, this Minkowski sum approach extends to type B. msc classes. 52B11, 52B12, 03G10, 06B10

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Partially supported by the French ANR grants CAPPS 17 CE40 0018 and CHARMS 19 CE40 0017.

Contents

Introduction 3

Part I.

Type A shard polytopes

I.1. Preliminaries 8

I.1.1. Fans and polytopes 8

I.1.2. Permutations and noncrossing arc diagrams 8

I.1.3. Weak order and canonical join and meet representations 9

I.1.4. Lattice quotients 10

I.1.5. Braid fan and permutahedron 13

I.1.6. Quotient fans and quotientopes 13

I.1.7. Shards 15

I.1.8. Minkowski sums of associahedra 16

I.2. Shard polytopes 17

I.2.1. Definition of shard polytopes 18

I.2.2. Basic geometric properties of shard polytopes 20

I.2.3. Normal fans of shard polytopes 22

I.2.4. Quotientopes from shard polytopes 24

I.2.5. A Minkowski identity on shard polytopes 26

I.3. Minkowski geometry of shard polytopes 27

I.3.1. Type cones and shard polytopes 28

I.3.1.1. Minkowski summands and type cone 28

I.3.1.2. Shard polytopes are indecomposable deformed permutahedra 29

I.3.2. Matroid polytopes and shard polytopes 31

I.3.2.1. Shard polytopes are matroid polytopes 31

I.3.2.2. Series-parallel matroid polytopes 31

I.3.3. Virtual deformed permutahedra and shard polytopes 32

I.3.3.1. From simplices to shard polytopes 34

I.3.3.2. From shard polytopes to simplices 37

I.3.3.3. From right hand sides to shard polytopes 39

I.3.3.4. From shard polytopes to right hand sides 40

I.3.4. PS-quotientopes via shard polytopes 41

I.3.5. Mixed volumes of shard polytopes 42

Part II.

Type B shard polytopes

II.1. Type B combinatorics and geometry 43

II.1.1. Type B permutations and noncrossing arc diagrams 43

II.1.2. Type B weak order and canonical join and meet representations 44

II.1.3. Type B lattice quotients 45

II.1.4. Type B Coxeter arrangement and permutahedron 47

II.1.5. Type B quotient fans and shards 49

II.2. Type B shard polytopes and quotientopes 51

II.2.1. Type B shard polytopes 51

II.2.2. Proof of Proposition 129 53

II.2.2.1. Separated B-arcs 54

II.2.2.2. Singular B-arcs 54

II.2.2.3. Overlapped B-arcs 54

Concluding remarks and further directions 59

Appendix A. Detailed descriptions of Minkowski sums of shard polytopes 64

A.1. Normal cones of vertices and edges of shard polytopes 64

A.2. Vertex and facet descriptions of Minkowski sums of shard polytopes 65

Introduction

Context. This paper deals with polytopal realizations of lattice quotients of the weak order of the symmetric group Sn and of the type B Coxeter group Sbn. The prototype is the classical Tamari

lattice on binary trees with n nodes [Tam51, Sta63], seen as the quotient of the weak order on permutations of [n]:= {1, . . . , n} by the sylvester congruence [LR98, HNT05]. Its Hasse diagram is the graph of the classical associahedron, which can be defined either as the convex hull of well-chosen points associated to all binary trees on n nodes [Lod04], or by deleting some well-chosen inequalities from the facet description of the permutahedron [SS93], or as the Minkowski sum of the faces of the standard simplex corresponding to all intervals of [n] [Pos09]. Other relevant examples of polytopal realizations of lattice quotients of the weak order on Sn include the cube

for the boolean lattice, C. Hohlweg and C. Lange’s associahedra [HL07, LP18] for N. Reading’s Cambrian lattices [Rea06,CP17], the permutreehedra for the permutree lattices [PP18], the brick polytopes [PS12] for the increasing flip lattice on acyclic twists [Pil18], Minkowski sums of opposite associahedra for the rotation lattice on diagonal rectangulations [LR12, Gir12], etc. In [PS19], V. Pilaud and F. Santos actually managed to construct polytopes realizing the quotient fan defined by N. Reading [Rea05] for an arbitrary lattice congruence of the weak order on Sn. In this

paper, we propose an alternative approach to construct polytopal realizations of this quotient fan, using Minkowski sums of elementary polytopes called shard polytopes. Besides containing and explaining the construction of [PS19], the motivation for this new construction is the possibility to extend it to lattice quotients of the poset of regions of hyperplane arrangements beyond the braid arrangement. Indeed, the combinatorics and the geometry of the poset of regions of a hyperplane arrangement are strongly tied, giving rise to a natural geometric realization of lattice quotients of tight arrangements via polyhedral fans. We refer to the recent surveys of N. Reading [Rea16b,

Rea16a] for an introduction to the topic. However, no general polytopal realization is known. We achieve the first step in this perspective by constructing quotientopes for any lattice quotient of the weak order of the type B Coxeter group.

Lattice congruences and arc ideals. Recall that a congruence of a finite lattice L is an equivalence relation compatible with the meet and join operations on L. A congruence of L is completely determined by the join-irreducibles of L it contracts (i.e. that are not minimal in their congruence classes). One says that j forces j0 if every congruence that contracts j also contracts j0, and this relation is acyclic when L is congruence uniform. The set of congruences of L ordered by refinement is a lattice isomorphic to the lattice of upper ideals of the forcing relation on join-irreducibles of L.

Specializing these observations provides a convenient and powerful combinatorial model to manipulate lattice congruences of the weak order [Rea15]. The join-irreducible elements of the weak order are the permutations with a single descent. Such a permutation σ can be naturally encoded by a quadruple (a, b, A, B) with a < b and A t B = ]a, b[, where A and B respectively record the values of ]a, b[ that appear before or after the unique descent ba in σ. This quadruple can be represented by a curve, calledarc, wiggling around points on the horizontal axis, joining a to b while passing above the points of A and below the points of B. The forcing relation on join-irreducible permutations translates to a simple forcing relation on arcs. Each lattice congruence ≡ of the weak order thus corresponds to an upper ideal A≡ of the forcing order among arcs, called

arc idealof ≡.

Shards, quotient fans, and quotientopes. Geometrically, the arcs correspond to pieces of hyperplanes, calledshards, that partition the braid arrangement. Namely, the arc α:= (a, b, A, B) corresponds to the shard S(α) defined as the piece of the hyperplane xa = xb defined by the

inequalities xa0≤ x_a= x_b≤ x_b0 for all a0 ∈ A and b0 ∈ B. In [Rea05], N. Reading proved that each lattice congruence ≡ of the weak order defines a complete fan F≡, calledquotient fan, whose

dual graph is the Hasse diagram of the lattice quotient Sn/≡. The chambers of F≡ can be seen

either by glueing together the chambers of the braid fan that belong to the same congruence class, or as the connected components of the complement of the union of the shards S(α) for all arcs α

in the ideal A≡. In [PS19], V. Pilaud and F. Santos showed that this quotient fan is the normal

fan of a polytope, calledquotientope. These realizations were obtained by a careful but slightly obscur choice of right-hand sides defining an inequality normal to each ray of the braid fan. In this paper, we propose an alternative approach to construct polytopal realizations of the quotient fan, with several advantages discussed below.

Minkowski sums of associahedra. Our realizations are obtained as Minkowski sums of ele-mentary pieces. To illustrate the idea, let us start with a simple construction. For any arc α, denote by Aα the arc ideal generated by α. The corresponding congruence ≡α is a

Cam-brian congruence [Rea06] and the corresponding quotient fan Fα is a Cambrian fan of [RS09].

It is the normal fan of the α-associahedron Assoα of [HL07]. (Strictly speaking, the setting

of [Rea06,RS09, HL07, HLT11] is slightly different: it is valid for arbitrary Coxeter groups, but is defined in full dimension, so that we should assume that the endpoints of α are 1 and n. Their constructions however extend straightforward to arbitrary arcs.) The motivating observation of this paper is the following statement.

Theorem 1. Consider an arbitrary congruence ≡ of the weak order, and let α1, . . . , αp denote

the arcs generating the ideal A≡. Then the quotient fan F≡ is

• the common refinement of the Cambrian fans Fα1, . . . , Fαp, and

• the normal fan of the Minkowski sum of the associahedra Assoα1, . . . , Assoαp.

Note that this observation was already made for certain specific quotients (e.g. for the Bax-ter congruence corresponding to diagonal rectangulations [LR12] or for intersections of essential Cambrian congruences [CP17]) but it was never exploited to realize quotient fans of arbitrary lattice congruences. In contrast to the intricate construction of [PS19], the simple construction of Theorem 1 has the advantage to transfer all the geometric difficulty into the construction of the α-associahedra, which was already done in [HL07]. Intuitively, each αi-associahedron Assoαi is responsible for the shards of the ideal Aαi to appear in the normal fan of the Minkowski sum. The main idea of this paper is to push this idea further. We already mentioned that the classical associahedron decomposes into the Minkowski sum of faces of the standard simplex corresponding to the intervals of [n] [Pos09]. In general, the α-associahedron can be decomposed further into Minkowski sums of more elementary pieces. These pieces are the central topic of this paper. Shard polytopes. For an arc α:= (a, b, A, B), we define theshard polytopeSP(α) as the convex hull of all vectors with coordinates in {−1, 0, 1} where the 1’s and −1’s alternate (starting with a 1 and ending with a −1), and the 1’s appear at some positions in {a} ∪ A while the −1’s appear at some positions in B ∪ {b}. The family of shard polytopes is interesting by itself: for instance, similarly to the families of permutahedra or associahedra, any face of a shard polytope is a Cartesian product of shard polytopes. But the crucial property of shard polytopes is the following.

Theorem 2. For any arc α, the union of the walls of the normal fan of the shard polytope SP(α) contains the shard S(α) and is contained in the union of the shards S(α0) for all arcs α0 forcing α. This property enables to construct quotientopes as Minkowski sums of shard polytopes. The idea now is that each shard polytope SP(α) will be responsible for the shard S(α) to appear in the normal fan of the Minkowski sum, without introducing unwanted walls.

Corollary 3. For any lattice congruence ≡ of the weak order and any positive coefficients sα> 0

for α ∈ A≡, the quotient fan F≡is the normal fan of the Minkowski sum SP(A≡):=Pα∈A≡sαSP(α) of the shard polytopes SP(α) of all arcs α ∈ A≡.

Already setting the coefficients sα= 1, this construction recovers relevant realizations of specific

quotient fans mentioned above. For the sylvester congruence, all shard polytopes are faces of the standard simplex and the Minkowski sum SP(A≡) is the classical associahedron of [SS93,

Lod04, Pos09]. More generally, for the α-Cambrian congruence, the Minkowski sum SP(Aα) is

the α-associahedron of [HL07]. In contrast, we show that the standard permutahedron is not a Minkowski sum of dilated shard polytopes, although other realizations of the braid fan are.

Type cones and shard polytopes. Pursueing the quest for elementary Minkowski summands, one could ask whether shard polytopes can be further decomposed into even simpler pieces. How-ever, we show that shard polytopes are elementary geometric and combinatorial objects.

Proposition 4. For any arc α, the shard polytope SP(α) is Minkowski indecomposable.

This statement can be rephrased in the realization spaces of the quotient fans. For a fan F , the space of all polytopes whose normal fan coarsens the fan F is a cone under Minkowski addition, called (closed) type coneby P. McMullen [McM73] ordeformation coneby A. Postnikov [Pos09,

PRW08]. For instance, the type cone of the braid fan is isomorphic to the classical space of submodular functions. Corollary3 and Proposition4affirm that for each arc α ∈ A≡, the shard

polytope SP(α) is a representative of a ray of the type cone of the quotient fan F≡. Type cones of

Cambrian fans have recently received particular attention with the works of [BMDM+_18,PPPP19,

BS18, JLS20]. Combining the results of these papers with Corollary 3 and Proposition 4 imply that shard polytopes can be interpreted as Newton polytopes of F -polynomials of cluster variables of acyclic type A cluster algebras [FZ02,FZ03,FZ07,BMDM+_{18], and brick polytope summands} of certain sorting networks [PS12,BS18,JLS20]. We are not aware that our vertex description in terms of alternating vectors had been observed earlier for these polytopes.

Matroid polytopes and shard polytopes. It is not difficult to derive from the definition that (up to a simple translation) shard polytopes are matroid polytopes, i.e. convex hull of the characteristic vectors of all bases of a matroid. It turns out that the resulting matroids actually belong to the relevant specific class of series-parallel matroids, defined as the graphical matroids of series-parallel graphs. More precisely, we explicitly define a graph Γαfor each arc α which yield

the following statement.

Proposition 5. For any arc α:_{= (a, b, A, B), the matroid polytope of the series-parallel graph Γα}

is the translated shard polytope−SP(α)→ := SP(α) + 1B∪{b}.

Series-parallel matroids are an important well known family [Oxl11, Sect. 5.4], and their matroid polytopes have been studied because of their extremal properties in the context of subdivisions arising from tropical linear spaces [Spe08,Spe09].

Signed Minkowski sums of simplices and of shard polytopes. As their normal fans coarsen the braid fan, shard polytopes belong to the class of deformed permutahedra studied in [Pos09,

PRW08] (we prefer the name “deformed permutahedra” rather than “generalized permutahedra” as there are many generalizations of permutahedra). It thus follows from [ABD10] that they decompose uniquely as a signed Minkowski sum of faces of the standard simplex. As a consequence of Proposition5, the coefficients in this decomposition can be expressed as signed beta invariants of the graphical matroid of Γα, which can be rewritten as follows. For I, J ⊆ [n] of cardinality at

least 2, we write I_{ J when {min J, max J} ⊆ ] min I, max I[ 4 I and ] min J, max J[ ∩ I ⊆ J.} Proposition 6. For any arc α:= (a, b, A, B), the translated shard polytope−SP(α)→ := SP(α)+1B∪{b}

decomposes as

−→

SP(α) =X

J

(−1)|J∩(B∪{a,b})|4J

where the sum is indexed by all subsets J ⊆ [a, b] such that |J | ≥ 2 and (A ∪ {a, b})_{ J.}

Conversely, we show that the faces of the standard simplex also decompose uniquely as a signed Minkowski sum of translated shard polytopes as follows.

Proposition 7. For any subset J ⊆ [n] such that |J | ≥ 2, the face 4J of the standard simplex

decomposes as

4J =

X

α :=(a,b,A,B)

(−1)|{a,b}∩{min J,max J}|−SP(α)→

where the sum is indexed by all arcs α:= (a, b, A, B) such that J_{ (A ∪ {a, b}).}

Proposition 8. Any deformed permutahedron has a unique decomposition as a Minkowski sum and difference of dilated shard polytopes (up to translation).

In other words, Propositions6and7give the exchange matrices between the two linear bases of the type cone of the braid arrangement given by the set of faces of the standard simplex and the set of translated shard polytopes. From Propositions6 and7, one can also deduce the matrices that transform the right hand sides of deformed permutahedra into their shard polytope coefficients, and vice versa.

PS-quotientopes revisited. These parametrizations of deformed permutahedra enable us in particular to prove that our construction recovers and explains that of V. Pilaud and F. San-tos [PS19]. Indeed, the quotientopes of [PS19] are parametrized by so-called forcing dominant functions f : 2[n]→ R>0. We show that, regardless of the forcing dominant function used in the

construction, the resulting quotientope can be obtained with the construction of Corollary3. Proposition 9. Any quotientope of [PS19] is a Minkowski sum of dilated shard polytopes (up to translation).

Mixed volumes of shard polytopes. Applying the machinery of A. Postnikov [Pos09], Propo-sition6yields summation formulas for the volume and mixed volumes [SY93] of shard polytopes. These formulas are based on the dragon marriage condition of [Pos09] to express mixed volumes of faces of the standard simplex.

Theorem 10. For any arcs α1, . . . , αn−1∈ An, the mixed volume of SP(α1), . . . , SP(αn−1) is

Vol(SP(α1), . . . , SP(αn−1)) = 1 (n − 1)! X J1,...,Jn−1 (−1)|J1|A1+···+|Jn−1|An−1_,

summing over all collections (J1, . . . , Jn−1) ∈ [n]_≥2

n−1

verifying (Ai∪{ai, bi})Jifor all i ∈ [n−1],

and such that J1, . . . , Jn−1 satisfies the dragon marriage condition of [Pos09].

Type B quotientopes. The second part of this paper is devoted to polytopal realizations of lattice quotients of the weak order of the type B Coxeter group. The prototype example here is the cyclohedron [BT94,Sim03,HL07]. In contrast to type A, no systematic construction of type B quotientopes was known so far.

The classical tool to manipulate type B objects is folding type A objects by central symmetry: the elements of the type Bn Coxeter group are commonly represented by signed permutations of

size n or centrally symmetric permutations of [±n]:= {−n, . . . , −1, 1, . . . , n}; the type Bn Coxeter

arrangement is the section of the type A2n−1 Coxeter arrangement by the centrally symmetric

space; and the n-dimensional cyclohedron is obtained from a (2n − 1)-dimensional associahe-dron Assoαfor some centrally symmetric arc α by a suitable projection ρb.

This folding procedure gives combinatorial and geometric models for lattice congruences of the type B weak order. Namely, the join-irreducible elements of Bn are in bijection with B-arcs

(i.e. centrally symmetric A-arcs or centrally symmetric pairs of non-crossing A-arcs on [±n]) and with B-shards (i.e. intersections of A-shards with the centrally symmetric space). Each lattice congruence ≡b _{of the type B then corresponds to an upper ideal A}b

≡b of the forcing order among B-arcs. It was also proved in [Rea05] that any lattice congruence ≡b _{of the type B weak order}

defines a quotient fan Fb

≡b, whose chambers can be constructed either by glueing together the chambers of the type B Coxeter fan that belong to the same congruence class, or as the connected components of the complement of the union of the B-shards for all B-arcs in the ideal Ab

≡b. Consider a type B congruence ≡b_{whose B-arc ideal A}b

≡b forms the A-arc ideal A≡ of a type A

congruence ≡. Then the quotient fan Fb

≡b is just the section of the quotient fan F≡ with the

centrally symmetric space. It can thus be realized by the image of any quotientope realizing F≡

under the projection map ρb_{, for instance by a Minkowski sum of cyclohedra of [HL07]. This is}

However, the main difficulty at this point is that, since the forcing order among B-arcs slightly differs from the forcing order among the corresponding A-arcs, some B-arc ideals do not form A-arc ideals. Already in type B2, some quotient fans cannot be realized as projections of type A

quotientopes, or as Minkowski sums of cyclohedra.

At this point, we need the fine granularity of shard polytopes. We define the shard poly-topeSP(β) of a B-arc β as the image of the shard polytope SP(α) of the corresponding A-arc α under the projection ρb_{. Again, the crucial property of type B shard polytopes is the following.}

Proposition 11. For any B-arc β, the union of the walls of the normal fan of the shard poly-tope SP(β) contains the shard S(β) and is contained in the union of the shards S(β0) for β ≺ β0.

This property provides the first proof that all type B quotient fans are polytopal.

Corollary 12. For any lattice congruence ≡b _{of the type B weak order and any positive}

co-efficients sβ > 0 for β ∈ Ab≡b, the quotient fan F≡bb is the normal fan of the Minkowski sum SP(Ab

≡b):= P

β∈AbsβSP(β) of the shard polytopes SP(β) of all B-arcs β ∈ Ab≡b.

Again, specializing to coefficients sβ = 1, we recover the cyclohedra of [HL07]. As in type A,

we believe that type B shard polytopes are elementary polytopes.

Conjecture 13. For any B-arc β, the shard polytope SP(β) is Minkowski indecomposable. We have checked this conjecture experimentally for all type Bnshard polytopes for small values

of n. However, the simple Minkowski indecomposability criterion that we use to prove Proposi-tion4 fails in type B and the proof of Conjecture13would require a much finer understanding of the facets of the type B shard polytopes.

If Conjecture13 holds, then the work of [BMDM+18,PPPP19,BS18,JLS20] implies that for any type B arc β which appears in the B-arc ideal of some type B Cambrian congruence, the shard polytope SP(β) is the Newton polytope of the F -polynomial of some type B cluster vari-able [FZ02,FZ03,FZ07], or equivalently some brick polytope summand of some type B subword complex [PS15,BS18,JLS20]. However, in contrast to the type A situation, there are B-arcs which do not belong to the B-arc ideal of any type B Cambrian congruence. We call themunsortable

B-arcs as the corresponding join-irreducible elements are not c-sortable for any Coxeter element c. We are not aware that the shard polytopes of these unsortable B-arcs appear in any other related construction.

Finally, as in type A, the type B shard polytopes forms a linear basis of the type cone of the type B Coxeter arrangement.

Proposition 14. Any type B deformed permutahedron has a unique decomposition as a Minkowski sum and difference of dilated type B shard polytopes (up to translation).

Note that this solves the problem posed in [ACEP20, Qu. 9.3] to find an explicit natural basis for the type cone of type B permutahedra. An interesting question that deserves further study is to describe the matrices that transform the right hand sides of type B deformed permutahedra into their type B shard polytope coefficients, and vice versa.

Further directions. We conclude the paper with a discussion on the existence of shard polytopes for arbitrary hyperplane arrangements. We show in particular, that they exist in type I2(n) for

any n, and discuss some progress on supersolvable arrangements. Let us conclude this introduction by mentioning that the existence of shard polytopes would have the following two consequences. Proposition 15. Consider a hyperplane arrangement H with a base region B such that the poset of regions PR(H, B) is a lattice, and a lattice congruence ≡ of PR(H, B) with shard ideal S≡. If

each shard S of S≡ admits a shard polytope PS, then the quotient fan F≡ is the normal fan of the

Minkowski sumP

S∈S≡PS.

Proposition 16. Consider a simplicial arrangement H and a base region B such that any shard S admits a shard polytope PS. Then any polytope in the deformation cone of the zonotope of H has

a unique decomposition as a Minkowski sum and difference of dilated shard polytopes PS for all

Part I. Type A shard polytopes

I.1. Preliminaries

We start with preliminaries on combinatorial and geometric properties of lattice quotients of the weak order on permutations. The presentation borrows from the papers [PS19, Pil19] and we reproduce here some of their pictures. Some combinatorial and lattice theoretic aspects pre-sented here (SectionsI.1.2andI.1.3) are not strictly needed for the geometric discussion of PartI, but will be useful to introduce analogous properties in type B in Part II. Throughout the pa-per, n is a fixed integer, and we use the notations [n]:= {1, . . . , n} as well as [a, b]:= {a, . . . , b} and ]a, b[:= {a + 1, . . . , b − 1} for two integers 1 ≤ a < b ≤ n.

I.1.1. Fans and polytopes. We start with basic notions of polyhedral geometry. We refer to G. Ziegler’s classic textbook [Zie98].

A hyperplane H ⊂ Rn is a supporting hyperplane _{of a set X ⊂ R}n _{if H ∩ X 6= ∅ and X is}

contained in one of the two closed half-spaces of Rn defined by H.

A (polyhedral) cone _{is a subset of R}n defined equivalently as the positive span of finitely many vectors or as the intersection of finitely many closed linear halfspaces. Its faces are its intersections with its supporting linear hyperplanes, and itsrays(resp.facets) are its dimension 1 (resp. codimension 1) faces. A (polyhedral)fan F is a collection of cones which are closed under faces (if C ∈ F and F is a face of C, then F ∈ F ) and intersect properly (if C, C0 ∈ F , then C ∩ C0 is a face of both C and C0). The chambers (resp. walls, resp. rays) of F are its codimension 0 (resp. codimension 1, resp. dimension 1) cones.

Apolytope_{is a subset of R}n _{defined equivalently as the convex hull of finitely many points or as}

a bounded intersection of finitely many closed affine halfspaces. Itsfacesare its intersections with its supporting affine hyperplanes, and its vertices (resp. edges, resp. facets) are its dimension 0 (resp. dimension 1, codimension 1) faces. Thenormal coneof a face F of a polytope P is the cone generated by the outer normal vectors of the facets of P containing F. Thenormal fanof P is the fan formed by the normal cones of all faces of P.

TheMinkowski sum_{of two polytopes P, Q ⊂ R}nis the polytope P+Q:= {p + q | p ∈ P, q ∈ Q}. For any r ∈ Rn, the face maximizing the direction r on P + Q is the Minkowski sum of the faces maximizing the direction r on P and Q. Therefore,

• the normal fan of P + Q is the common refinement of the normal fans of P and Q, • the vertex of P + Q maximizing a generic r is the sum of vertices of P and Q maximizing r, • the facet of P + Q maximizing a ray r is defined by h r | x i = max

p∈P h r | p i + maxq∈Qh r | q i.

Further notions on polyhedral geometry and in particular on properties of Minkowski sums will be recalled along the text when needed, in particular weak Minkowski decompositions and type cones in SectionI.3.1, virtual polytopes in SectionI.3.3, and mixed volumes in SectionI.3.2. I.1.2. Permutations and noncrossing arc diagrams. We now briefly recall the bijection be-tween permutations and noncrossing arc diagrams developed by N. Reading in [Rea15]. The lattice theoretic interpretation of this bijection is presented in SectionI.1.3

An arc is a quadruple (a, b, A, B) consisting of two integers 1 ≤ a < b ≤ n and a parti-tion A t B = ]a, b[. This arc can be visually represented as an x-monotone continuous curve wiggling around the horizontal axis, with endpoints a and b, and passing above the points of A and below the points of B. For instance, represents the arc (1, 4, {2}, {3}). Note that the same information could be recorded in a more compact way, for instance by forgetting ei-ther B = ]a, b[ r A, or a = min(A t B) − 1 and b = max(A t B) + 1. Nevertheless, it is somehow convenient in this paper to keep the complete notation (a, b, A, B). We denote the set of all arcs by An := {(a, b, A, B) | 1 ≤ a < b ≤ n and A t B = ]a, b[}. Note that |An| = 2n− n − 1.

We say that two arcs (a, b, A, B) and (a0, b0, A0, B0)crossif the interior of the two curves repre-senting these arcs intersect in their interior, that is, if both A ∩ ({a0, b0} ∪ B0_{) and ({a, b} ∪ B) ∩ A}0

are non-empty. A noncrossing arc diagram is a collection of arcs of An where any two arcs do

1 2 3 4 5 6 7 7 6 4 2 1 3 5 1 2 3 4 5 6 7 7 6 4 2 1 3 5 1 2 3 4 5 6 7 7 6 4 2 1 3 5 1 2 3 4 5 6 7 7 6 4 2 1 3 5

Figure 1. The noncrossing arc diagrams δ(σ) (bottom) and δ(σ) (top) for the four permutations σ = 2537146, 2531746, 2513746, and 2513476.

an arc can be the left endpoint of an other arc). See Figure 1 for examples of noncrossing arc diagrams.

Consider now the set Sn of permutations of [n]. Represent a permutation σ ∈ Sn by its

permutation tableformed by dots at coordinates (σi, i) for i ∈ [n]. (This unusual choice of

orien-tation fits with the existing constructions [LR98,HNT05, CP17, PP18, Pil19].) Draw a segment between any two consecutive dots (σi, i) and (σi+1, i + 1), colored blue for an ascent σi < σi+1

and red for a descent σi > σi+1. Then move all dots down to the horizontal axis, allowing the

segments to curve, but not to cross each other nor to pass through any dot, as illustrated in Figure1. The resulting set of blue (resp. red) arcs is a noncrossing arc diagram δ(σ) (resp. δ(σ)). Less visually and more formally, the blue diagram is δ(σ) = {α(σ, i) | σi< σi+1} where α(σ, i) is

the arc (σi, σi+1, {σj | j < i, σi< σj < σi+1} , {σj| j > i + 1, σi< σj < σi+1}) (resp. the red

dia-gram is δ(σ) = {α(σ, i) | σi> σi+1} where α(σ, i) is the arc (σi+1, σi, {σj | j < i, σi+1< σj < σi} ,

{σj| j > i + 1, σi+1 < σj< σi}). These maps were introduced by N. Reading in [Rea15], where

he proved the following statement.

Theorem 17 ([Rea15, Thm. 3.1]). The map δ (resp. δ) is a bijection from the permutations of Sn

to the noncrossing arc diagrams on An.

The reverse bijections δ−1and δ−1are explicitly described in [Rea15, Prop. 3.2]. Briefly speak-ing, consider the poset of connected components of D ordered by (the transitive closure of) the priority X < Y if there is an arc (a, b, A, B) ∈ D with A ∩ X 6= ∅ and a, b ∈ Y , or with a, b ∈ X and B ∩ Y 6= ∅. To obtain δ−1(D) (resp. δ−1(D)), choose the linear extension of this priority poset where ties are resolved by choosing first the leftmost (resp. rightmost) connected component, and order decreasingly (resp. increasingly) the values in each connected component. See Figure1. I.1.3. Weak order and canonical join and meet representations. Consider a finite lat-tice (L, ≤, ∧, ∨), i.e. a finite set L partially ordered by ≤ where each subset X of elements admits a meet V X (greatest lower bound) and a join W X (least upper bound). A join representa-tion of x ∈ L is a subset J ⊆ L such that x =W J. Such a representation is irredundant

if x 6= W J0 _{for every strict subset J}0

( J . The irredundant join representations of an ele-ment x ∈ L are ordered by containele-ment of the lower ideals of their eleele-ments, i.e. J ≤ J0 if and only if for any y ∈ J there exists y0 ∈ J0 _{such that y ≤ y}0 _{in L. When this order has a}

min-imal element, it is called the canonical join representation of x. All elements of the canonical join representation x = W J are then join-irreducible, i.e. cover a single element. A lattice is

Thm. 2.24], x ∨ z = y ∨ z =⇒ x ∨ z = (x ∧ y) ∨ z for any x, y, z ∈ L. Canonical meet representa-tions,meet-irreducible elementsandmeet-semidistributive latticesare defined dually. A lattice is

semidistributiveif it is both join- and meet-semidistributive.

We consider the weak order on permutations of Sn defined by σ ≤ τ ⇐⇒ inv(σ) ⊆ inv(τ )

where inv(σ):= {(σa, σb) | 1 ≤ a < b ≤ n and σa> σb} is the inversion setof the permutation σ.

A cover relation in the weak order corresponds to a swap of two letters at consecutive positions. See Figure4 for the Hasse diagram of the weak order on S4 and some geometric representations

recalled in Section I.1.5. The permutations covered by (resp. covering) a permutation σ ∈ Sn

in weak order correspond to the descents (resp. ascents) of σ. Hence, a permutation σ ∈ Sn

is join-irreducible (resp. meet-irreducible) in weak order if and only if it has a unique descent (resp. ascent).

The weak order on Sn is a semidistributive lattice, and the canonical join and meet

representa-tions of a permutation were described in [Rea15] as follows. Consider an arc α:_{= (a, b, A, B) ∈ An,}

where A:_{= {a1< · · · < ap} and B}:_{= {b1< · · · < bq}. We associate to the arc α the join-irreducible}

permutation λ(α):_{= [1, . . . , a − 1, a1, . . . , ap, b, a, b1, . . . , bq, b + 1, . . . , n] (resp. the meet-irreducible}

permutation λ(α):_{= [n, . . . , b + 1, bq, . . . , b1, a, b, ap, . . . , a1, a − 1, . . . , 1]). The canonical meet and}

join representations of σ are then given by their red and blue noncrossing arc diagrams, which provides a lattice theoretic interpretation of the bijections of Theorem17.

Theorem 18 ([Rea15, Thm. 2.4]). The canonical join and meet representations of a permutation σ are given by W {λ(α) | α ∈ δ(σ)} and Vλ(α)

α ∈ δ(σ) .

I.1.4. Lattice quotients. We now consider lattice congruences and lattice quotients of the weak order. For details, we refer to the thorough work of N. Reading, in particular the articles [Rea04,

Rea05,Rea06,Rea15] and the surveys [Rea12,Rea16b,Rea16a].

A lattice congruence of a lattice (L, ≤, ∧, ∨) is an equivalence relation on L that respects the meet and the join operations, i.e. such that x ≡ x0 and y ≡ y0 implies x ∧ y ≡ x0 ∧ y0

and x ∨ y ≡ x0∨ y0_{. Equivalently, the equivalence classes of ≡ are intervals of L, and the up and}

down maps π↑and π↓, respectively sending an element of L to the top and bottom elements of its

≡-equivalence class, are order-preserving. A lattice congruence ≡ defines a lattice quotientL/≡ on the congruence classes of ≡ where X ≤ Y if and only if there exist x ∈ X and y ∈ Y such that x ≤ y, and X ∧ Y (resp. X ∨ Y ) is the congruence class of x ∧ y (resp. x ∨ y) for any x ∈ X and y ∈ Y . Intuitively, the quotient L/≡ is obtained by contracting the equivalence classes of ≡ in the lattice L. More precisely, we say that an element x is contractedby ≡ if it is not minimal in its equivalence class of ≡, i.e. if x 6= π↓(x). As each class of ≡ is an interval of L,

it contains a unique uncontracted element, and the quotient L/≡ is isomorphic to the subposet of L induced by its uncontracted elements π↓(L).

Example 19. The prototype lattice congruence of the weak order is thesylvester congruence≡sylv

[LR98,HNT05]. Its congruence classes are the fibers of the binary search tree insertion algorithm, or equivalently the sets of linear extensions of binary trees (labeled in inorder and considered as posets oriented from bottom to top). It can also be seen as the transitive closure of the rewriting rule U ikV jW ≡sylvU kiV jW where i < j < k are letters and U, V, W are words on [n].

In other words, the uncontracted permutations in the sylvester congruence are those avoiding the pattern 231. The quotient of the weak order by the sylvester congruence is (isomorphic to) the classicalTamari lattice [Tam51], whose elements are the binary trees on n nodes and whose cover relations are rotations in binary trees. The sylvester congruence and the Tamari lattice are illustrated in Figure2 for n = 4. We will use the sylvester congruence and the Tamari lattice as a familiar example throughout the paper.

If a lattice L is semidistributive, then any lattice quotient L/≡ is also semidistributive. More-over, via the identification between ≡-classes and their minimal elements, the canonical join rep-resentations in the quotient L/≡ ' π↓(L) are precisely the canonical join representations of L

that only involve join-irreducibles of L uncontracted by ≡. We have seen in SectionI.1.3that the weak order on Sn is semidistributive, that its join-irreducibles correspond to arcs of An, and that

the canonical join representations of permutations correspond to noncrossing arc diagrams. This yields the following statement.

4321 4231 4312 3421 3412 2431 3241 4213 4132 1234 1324 1243 2134 2143 3124 2314 1342 1423 3142 2413 4123 1432 2341 3214

Figure 2. The quotient of the weak order by the sylvester congruence ≡sylv(left) is the Tamari

lattice (middle). The quotient fan Fsylvis the normal fan of J.-L. Loday’s associahedron (right).

Theorem 20 ([Rea15, Thm. 4.1]). For any lattice congruence ≡ of the weak order on Sn, the

set of join-irreducibles of Sn uncontracted by ≡ correspond to a set of arcs A≡, and the canonical

join representations in the lattice quotient Sn/≡ correspond to noncrossing arc diagrams using

only arcs of A≡.

Example 21. For the sylvester congruence ≡sylvof Example19, the uncontracted join-irreducibles

are given by the set Asylv= {(a, b, ]a, b[, ∅) | 1 ≤ a < b ≤ n} of up arcs, i.e. those which pass above

all dots in between their endpoints. Therefore, the sylvester congruence classes are in bijection with noncrossing arc diagrams with arcs in Asylv, also known as noncrossing partitions.

The set of all lattice congruences of a lattice L ordered by refinement is a lattice whose meet is the intersection of congruences and join is the transitive closure of union of congruences. In particular, for any join-irreducible element j of L, there is a unique minimal lattice congruence ≡j

contracting j, and any lattice congruence ≡ of L is the joinW

j≡jover all join-irreducible elements j

of L contracted by ≡. For two join-irreducible elements j, j0 of L, we say that j forces j0, and write j  j0, if every congruence that contracts j also contracts j0. The forcing relation is a preposet and thus defines a poset on its equivalence classes, called forcing poset of L. It follows that the lattice of congruences of L is isomorphic to the lattice of upper ideals on the forcing poset of L. Let us finally mention that a lattice is called congruence uniformwhen the forcing relation is a poset, so that the map j 7→ ≡j is a bijection between the join-irreducible elements of L and

that of the lattice of congruences of L.

The weak order is a congruence uniform lattice, and the forcing order on join-irreducibles can be described visually on arcs as follows. We say that an arc α:= (a, b, A, B) ∈ An forces an

arc α0 := (a0, b0, A0, B0) ∈ An, and we write α  α0, if a0 ≤ a < b ≤ b0 and A ⊆ A0 and B ⊆ B0.

Visually, α forces α0 if the endpoints of α are located in between those of α0 and α agrees with α0 in between its endpoints. We call arc poset the poset (An, ≺) of all arcs ordered by inverse

forcing (small elements are forced by big elements in this poset). The forcing relation and the arc poset on A4 are illustrated on Figure 3. We thus obtain the following description of the lattice

congruences of the weak order.

Theorem 22 ([Rea15, Thm. 4.4 & Coro. 4.5]). The map ≡ 7→ A≡ is a bijection between the

lattice congruences of the weak order on Sn and the upper ideals of the arc poset (An, ≺).

We callarc idealan upper ideal of the arc poset (An, ≺). In view of this statement, we make no

distinction between arc ideals and lattice congruences. When needed, we write ≡Afor the lattice

congruence of the weak order on Sn corresponding to an arc ideal A ⊆ An.

Example 23. For an arc α:= (a, b, A, B) ∈ An, we denote by Aα := {α0∈ An | α ≺ α0} the upper

a

a´

b

b´

Figure 3. The forcing relation among arcs (left) and the arc poset for n = 4 (right).

on Sn is called the α-Cambrian congruence, and the lattice quotient Sn/≡α is the α-Cambrian

lattice. It was introduced and extensively studied by N. Reading in [Rea06] (technically, it was slightly different: Cambrian lattices were introduced in the general context of finite Coxeter groups, and in type A they would only correspond to arcs with endpoints a = 1 and b = n). For instance, the sylvester congruence of Example19_{is the (1, n, ]1, n[, ∅)-Cambrian congruence, and}

the Tamari lattice is the (1, n, ]1, n[, ∅)-Cambrian lattice. The α-Cambrian congruence classes are fibers of the α-Cambrian tree insertion, or equivalently linear extensions of α-Cambrian trees, see [LP18, CP17, PP18] (again, the setting in these papers assumes that a = 1 and b = n, but there is no difficulty to adapt it to arbitrary α). Let us just say that an α-Cambrian tree is a tree on [a, b] such that the node j ∈ {a} ∪ A (resp. j ∈ B ∪ {b}) has one ancestor (resp. descendant) subtree and two descendant (resp. ancestor) subtrees, and i < j < k for any nodes i in the left descendant (resp. ancestor) subtree of j and k in the right descendant (resp. ancestor) subtree of j. The α-Cambrian congruence can also be seen as the transitive closure of the three rewriting rules U ijV ≡α U jiV for i < a or j > b, U ikV jW ≡α U kiV jW for i < j < k with j ∈ A,

and U jV ikW ≡αU jV kiW for i < j < k with j ∈ B. In particular, the uncontracted permutations

in the α-Cambrian congruence are those avoiding the consecutive patterns ji with i < j and i < a or j > b, and the patterns kji for i < j < k with j ∈ A and jik for i < j < k with j ∈ B.

Example 24. Let us gather some other relevant examples of lattice congruences of the weak order on Sn, as some of them will appear along this paper:

(1) the recoil congruence ≡rec is defined by the ideal Arec = {(i, i + 1, ∅, ∅) | i ∈ [n − 1]} of

basic arcs. It has a congruence class for each subset I ⊆ [n − 1] given by the permutations whose recoils (descents of the inverse) are at positions in I. It can also be seen as the tran-sitive closure of the rewriting rule U ijV ≡recU jiV for |i − j| > 1. The quotient Sn/≡rec

is the boolean lattice.

(2) for δ ∈ { , , , }n_{, the δ-permutree congruence ≡}

δ is defined by the ideal Aδ of

arcs which do not pass above the points j with δj ∈ { , } nor below the points j

with δj ∈ { , }. Its congruence classes correspond to δ-permutrees [PP18]. It can also

be seen as the transitive closure of the rewriting rules U ikV jW ≡δU kiV jW for i < j < k

with δj∈ { , } and U jV ikW ≡δ U jV kiW for i < j < k with δj ∈ { , }.

(3) theBaxter congruence≡Baxis defined by the ideal of arcs that do not cross the horizontal

axis, i.e. ABax= {(a, b, A, B) ∈ An| A = ∅ or B = ∅}. Its congruence classes correspond

to diagonal rectangulations [LR12] or equivalently pairs of twin binary trees [Gir12], which are counted by the Baxter numbers. It can also be seen as the transitive closure of the rewriting rule U jV i`W kX ≡BaxU jV `iW kX for i < j, k < `.

(4) for p ≥ 1, thep-recoil congruence≡p-rec is defined by the ideal of arcs of length at most p,

i.e. Ap-rec = {(a, b, A, B) ∈ An | b − a ≤ p}. Its congruence classes correspond to acyclic

orientations of the graph on [n] with edges (a, b) for |a − b| ≤ p. It can also be seen as the transitive closure of the rewriting rule U ijV ≡p-rec U jiV for |i−j| > p. See [Rea05,Pil18].

4321 4231 4312 3421 3412 2431 3241 4213 4132 1234 1324 1243 2134 2143 3124 2314 1342 1423 3142 2413 4123 1432 2341 3214 1234 2134 2314 3214 3124 1324 1243 1342 3142 3241 3421 4321 4312 ₃₄₁₂ 4132 1432 4123 1423 3412 4312 4321 ₃₄₂₁ 3142 3241 3214 1342 1432 1423 1243 ₁₂₃₄ 2134 1324 4123 4132 2314 3124 2143 2413 4213 2431 4231 2341

Figure 4. The Hasse diagram of the weak order on S4 (left) can be seen as the dual graph of

the braid fan F4(middle) or as the graph of the permutahedron Perm4(right).

(5) for p ≥ 1, the p-twist congruence≡p-twist is defined by the ideal of arcs passing below at

most p points, i.e. Ap-twist = {(a, b, A, B) ∈ An| |B| ≤ p}. Its congruence classes

corre-spond to certain acyclic pipe dreams [Pil18]. It can also be seen as the transitive closure of the rewriting rule U ikV1j1. . . VpjpW ≡p-twistU kiV1j1. . . VpjpW for i < j1, . . . , jp< k.

Remark 25. The Hasse diagram of a lattice quotient Sn/≡ is not always regular (i.e. of constant

degree). H. Hoang and T. M¨utze proved in [HM19] that it is regular if and only if all maximal arcs of Anr A≡ are of the form (a, b, ]a, b[, ∅) or (a, b, ∅, ]a, b[). This holds for instance for all

α-Cambrian congruences of [Rea06] and more generally for all permutree congruences of [PP18] I.1.5. Braid fan and permutahedron. We now switch to some geometric considerations on the weak order on Sn. Namely, its Hasse diagram can be seen geometrically as the dual graph of the

braid fan Fn, or as the graph of the permutahedron Permn defined below, oriented in the linear

direction γ:=P

i∈[n](2i − n − 1) ei = (−n + 1, −n + 3, . . . , n − 3, n − 1).

The braid arrangement is the set Hn of hyperplanes {x ∈ Rn | xa= xb} for 1 ≤ a < b ≤ n.

As all hyperplanes of Hn contain the line R1:= R(1, 1, . . . , 1), we restrict to the hyperplane

H:₌x ∈ Rn  P

i∈[n]xi = 0 . The hyperplanes of Hn divide H into chambers, which are the

maximal cones of a complete simplicial fan Fn, calledbraid fan. It has

• a chamber C(σ):_{= {x ∈ H | xσ}

1≤ xσ2 ≤ · · · ≤ xσn} for each permutation σ of Sn, • a ray C(R):=_{x ∈ H}

xr1 = · · · = xrp ≤ xs1 = · · · = xsn−p for each subset ∅ 6= R ( [n], where R = {r1, . . . , rp} and [n] r R = {s1, . . . , sn−p}. When needed, we use the

represen-tative vector r(R):= |R|1 − n1Rin C(R), where 1:=Pi∈[n]ei and 1R :=Pr∈Rer.

The chamber C(σ) has rays C(σ([k])) for k ∈ [n]. See Figures4(middle),5(left) and6(left) where the chambers are labeled in blue and the rays are labeled in red. Note that Fn has n! chambers,

n!(n − 1)/2 walls supported by n₂
hyperplanes, and 2n_{− 2 rays.}

Thepermutahedronis the polytope Permn defined equivalently as

• the convex hull of the pointsP

i∈[n]i eσi for all permutations σ ∈ Sn,

• the intersection of the hyperplane H:=_{x ∈ R}n

 P_i∈[n]xi= n+1₂
with the halfspaces

x ∈ Rn

 P_r∈Rxr≥ |R|+12
for all proper subsets ∅ 6= R ( [n],

• (a translate of) the Minkowski sum of all segments [ea, eb] for all 1 ≤ a < b ≤ n.

See Figure4(right). Note that Permn has n! vertices, n!(n − 1)/2 edges, and 2n− 2 facets. The

normal fan of the permutahedron Permn is the braid fan Fn.

I.1.6. Quotient fans and quotientopes. We now consider the geometry of lattice quotients of the weak order. First, lattice congruences naturally yield quotient fans described in the following statement. Although stated in the more general context of hyperplane arrangements in [Rea05] (see also [Rea16b]), we prefer to focus here on its simple version on the braid arrangement.

Theorem 26 ([Rea05]). Any lattice congruence ≡ of the weak order on Sn defines a complete

fan F≡, calledquotient fan, whose chambers are obtained by glueing together the chambers C(σ) of

the braid fan Fn corresponding to the permutations σ that belong to the same congruence class of ≡.

As it turns out, the quotient fans of all lattice congruences of the weak order are polytopal. Theorem 27 ([PS19]). For any lattice congruence ≡ of the weak order on Sn, the quotient fan F≡

is the normal fan of a polytope P≡, called quotientope.

By construction, the Hasse diagram of the quotient of the weak order by ≡ is given by the dual graph of the quotient fan F≡, or by the graph of the quotientope P≡, oriented in the direction γ.

Example 28. For the sylvester congruence ≡sylvof Examples19and21, the quotient fan Fsylvhas

• a chamber C(T ) = {x ∈ H | xa≤ xb if a is a descendant of b in T } for each binary tree T ,

• a ray C(I) for each proper interval I = [i, j] ( [n].

Figures 5 and 6 (right) illustrate the quotient fans Fsylv for n = 3 and n = 4. The quotient

fan Fsylvis the normal fan of the classical associahedron Asson defined equivalently as:

• the convex hull of the points P

j∈[n]`(T, j) r(T, j) ej for all binary trees T on n nodes,

where `(T, j) and r(T, j) respectively denote the numbers of leaves in the left and right subtrees of the node j of T (labeled in inorder), see [Lod04],

• the intersection of the hyperplane H with the halfspacesx ∈ Rn 

P

a≤i≤bxi≥ b−a+2₂

for all intervals 1 ≤ a ≤ b ≤ n, see [SS93],

• (a translate of) the Minkowski sum of 4[a,b]for all intervals 1 ≤ a ≤ b ≤ n, where for I ⊆ [n],

4I := conv {ei| i ∈ I} is the face of the standard simplex 4[n] labeled by I, see [Pos09].

Figure2 (right) shows the associahedron Asso4.

Example 29. Consider the α-Cambrian congruence of an arc α:= (a, b, A, B) defined in Exam-ple23. The quotient fan is theα-Cambrian fanFα. It has lineality space generated by (ei)i /∈[a,b],

and its section by R[a,b] _has

• a chamber C(T ) = {x ∈ H | xa ≤ xb if a is a descendant of b in T } for each α-Cambrian

tree T (see [LP18,CP17,PP18] or the brief description in Example23),

• a ray C(R) for each proper subset ∅ 6= R ( [a, b] such that for all a ≤ i < j < k ≤ b, if i, k ∈ R then j ∈ R ∪ B, and if i, k /∈ R then j /∈ R ∩ B.

The quotient fan Fαis the normal fan of C. Hohlweg and C. Lange’sα-associahedronAssoα[HL07]

defined equivalently as:

• the convex hull of the points P

j∈[a,b]HL(T, j) ej for all α-Cambrian trees T , where

HL(T, j) = `(T, j) r(T, j) (resp. HL(T, j) = b − a + 2 − `(T, j) r(T, j)), where `(T, j) and r(T, j) respectively denote the number of leaves in the left and right descendant (resp. ancestor) subtrees of the node j of T if j ∈ {a} ∪ A (resp. if j ∈ B ∪ {b}),

• the intersecton of the hyperplane x ∈ Rn  

P

i∈[n]xi = b−a+2₂
with the halfspaces

x ∈ Rn 

P

r∈Rxr≥ |R|+1₂
for all rays R described above.

(Again, the setting in [HL07] assumes that a = 1 and b = n, but there is no difficulty to adapt it to an arbitrary arc α.) See Figures11and12for the 2- and 3-dimensional associahedra Assoα.

Example 30. The quotient fans of the congruences of Example24are realized by: (1) the parallelotope P

i∈[n−1][ei, ei+1] for the recoil congruence,

(2) the δ-permutreehedron for the δ-permutree congruence [PP18],

(3) the Minkowski sum of Asson and −Asson for the Baxter congruence [LR12],

(4) the graphical zonotope P

|a−b|≤p[ea, ep] for the p-recoil congruence [Pil18],

(5) the brick polytope for the p-twist congruence [PS12,Pil18].

For an arc ideal A ⊆ An, we denote by FA the quotient fan F≡A of the corresponding lattice congruence ≡Avia the bijection of Theorem22. We will use the following characterization of the

rays of the quotient fan FA, proved for instance in [APR20, Sect. 3.1]. Note that it fits with the

descriptions of the rays of the quotient fans of the sylvester and Cambrian congruences given in Examples28and29.

123

321

213

132

231

312

x

>x

x

>x

x

>x

13

2

23

3

12

1

Figure 5. The braid fan F3(left), the corresponding shards (middle), and the quotient fan given

by the sylvester congruence ≡sylv (right).

Lemma 31 ([APR20, Sect. 3.1]). For any arc ideal A ⊆ An and any proper subset ∅ 6= R ( [n],

the ray C(R) of the braid fan Fn is also a ray of the quotient fan FA if and only for every

1 ≤ a < b ≤ n, we have (a, b, ∅, ]a, b[) ∈ A if a, b ∈ R and ]a, b[ ∩ R = ∅, and (a, b, ]a, b[, ∅) ∈ A if a, b /∈ R and ]a, b[ ⊆ R.

I.1.7. Shards. An alternative description of the quotient fan F≡defined in Theorem26is given by

its walls, each of which can be seen as the union of some preserved walls of the braid arrangement. The conditions in the definition of lattice congruences impose strong constraints on the set of preserved walls: deleting some walls forces to delete others. Shards were introduced by N. Reading in [Rea03] (see also [Rea16b,Rea16a]) to understand the possible sets of preserved walls.

For any arc α:= (a, b, A, B), theshardS(α) = S(a, b, A, B) is the cone

S(a, b, A, B):_{= {x ∈ H | xa} = xb, xa≥ xa0 for all a0∈ A, x_a≤ x_b0 for all b0∈ B} . We denote by Sn := {S(α) | α ∈ An} the set of all shards of Hn. Note that |Sn| = |An| = 2n−n−1

is the number 2n_{− 2 of rays of F}

n minus the dimension n − 1, see Lemma146.

Figures 5 and 6 illustrate the braid fans Fn and their shards Sn when n = 3 and n = 4

respectively. As the 3-dimensional fan F4 is difficult to visualize (as in Figure 4(middle)), we

use another classical representation in Figure 6(left): we intersect F4 with a unit sphere and we

stereographically project the resulting arrangement of great circles from the pole 4321 to the plane. Each circle then corresponds to a hyperplane xa = xbwith a < b, separating a disk where xa < xb

from an unbounded region where xa > xb. In both Figures 5 and 6, the left picture shows the

braid fan Fn (where chambers are labeled with blue permutations of [n] and rays are labeled with

red proper subsets of [n]), the middle picture shows the shards Sn (labeled by arcs), and the right

picture represents the quotient fan Fsylvof the sylvester congruence.

It turns out that the shards are precisely the pieces of the hyperplanes of Hn needed to delimit

the cones of the quotient fans. For A ⊆ An, we denote by SA := {S(α) | α ∈ A} the set of shards

corresponding to the arcs of A.

Theorem 32 ([Rea16a, Sect. 10.5]). For any arc ideal A ⊆ An, the union of the walls of FA is

the union of the shards of SA.

Example 33. Following Examples19,21and28, Figures5and6represent the quotient fans Fsylv

corresponding to the sylvester congruences ≡sylvon S3 and S4 respectively. It is obtained

• either by glueing the chambers C(σ) of the permutations σ in the same sylvester class, • or by cutting the space with the shards of Ssylv= {S(a, b, ]a, b[, ∅) | 1 ≤ a < b ≤ 4}.

x₁<x₂ x₃<x₄ x2<x3 x1>x3 x2>x4 x1>x4 4 34 234 24 3 134 3421 3412 4231 4312 2431 4213 3241 4132

Figure 6. A stereographic projection of the braid fan F4 (left) from the pole 4321, the

corre-sponding shards (middle), and the quotient fan given by the sylvester congruence ≡sylv (right).

Finally, note that the shards and the forcing order among them can also be constructed in a purely geometrical way. Namely, for any codimension 2 face F of the braid fan Fn, consider the

subarrangement HF

n of Hn formed by the hyperplanes containing F , and call F -basic the two

hyperplanes delimiting the region of HF

n containing the fundamental chamber C(1 . . . n) of Fn.

Cut the non-F -basic hyperplanes by the F -basic hyperplanes for each codimension 2 face F . The resulting connected components are the (open) shards of Sn. The forcing relation is obtained as

the transitive closure of the cutting relation: S cuts S0if there is a codimension 2 face F contained in S and S0 such that the supporting hyperplane of S is F -basic while that of S0 is not. Note that the cutting relation is weaker than the forcing relation: translated back on arcs, α cuts α0 if α forces α0 and α and α0 share an endpoint. See [Rea16b] for more details.

I.1.8. Minkowski sums of associahedra. To conclude this preliminary section, we show al-ternative polytopal realizations of the quotient fans using Minkowski sums of C. Hohlweg and C. Lange’s associahedra [HL07]. This construction was already used for certain specific quotients, e.g. for the Baxter congruence corresponding to diagonal rectangulations [LR12] or for intersec-tions of essential Cambrian congruences [CP17]. Our constructions in SectionI.2.4will be based on similar ideas.

Recall from Examples23 and29 that for an arc α ∈ An, we denote by Aα the upper ideal of

the arc poset (An, ≺) generated by α, by ≡αthe α-Cambrian congruence, by Fαthe α-Cambrian

fan, and by Assoα the α-associahedron. Our main tool is the following observation.

Lemma 34. For any arc ideal A ⊆ An with minimal elements α1, . . . , αp, the lattice

congru-ence ≡A is the intersection of the Cambrian congruences ≡α1, . . . , ≡αp.

Proof. The arc ideal A is generated by its minimal elements α1, . . . , αp, thus it is the union

of the principal ideals Aα1, . . . , Aαp. Therefore, the congruence ≡A is the intersection of the

congruences ≡α1, . . . , ≡αp. 

Lemma 34has the following direct combinatorial and geometric consequences. All these con-sequences where already observed in [CP17, arXiv version, Sect. 2.3] for intersection of essential Cambrian congruences (i.e. with a = 1 and b = n), but the observation of Lemma34was missing. Corollary 35. For any arc ideal A ⊆ Anwith minimal elements α1, . . . , αp, each congruence class

of ≡Ais represented by a p-tuple of α1-, . . . , αp-Cambrian trees with a common linear extension.

Proof. By Lemma 34, the ≡A-congruence classes are precisely the non-empty intersections of

≡α1-, . . . , ≡αp-congruence classes. Each αi-congruence class is the set of linear extensions of an

Figure 7. The quotient fan FBaxof the Baxter congruence is the normal fan of the Minkowski

sum Asso_{(1,n,[n],∅)}+ Asso_{(1,n,∅,[n])} of two opposite associahedra [LR12].

Corollary 36. For any arc ideal A ⊆ An with minimal elements α1, . . . , αp, the quotient fan FA

is the common refinement of the quotient fans Fα1, . . . , Fαp.

Proof. The union of the walls of FAis the union of the shards of SA, thus the union of the shards

of Sα1, . . . , Sαp, thus the union of the walls of Fα1, . . . , Fαp. It follows that the quotient fan FAis the common refinement of the quotient fans Fα1, . . . , Fαp.  Corollary 37. For any arc ideal A ⊆ An with minimal elements α1, . . . , αp, the quotient fan FA

is the normal fan of the Minkowski sum of the associahedra Assoα1, . . . , Assoαk.

Proof. For any arc α ∈ An, the quotient fan Fα is the normal fan of the α-associahedron Assoα.

The statement thus follows from Corollary 36and the fact that the normal fan of a Minkowski sum is the common refinement of the the normal fans of its summands. _ Example 38. Consider the Baxter congruence ≡Bax of Example 24(3). As already observed

in Example 30 and [LR12], the quotient fan FBax is the normal fan of the Minkowski sum of

J.-L. Loday’s associahedron [Lod04] and its opposite, see Figure7. This was extended in [CP17] to arbitrary pairs of opposite associahedra of C. Hohlweg and C. Lange [HL07], and even to arbitrary intersections of essential Cambrian congruences (i.e. with a = 1 and b = n).

I.2. Shard polytopes

In this section, we construct alternative realizations of the quotient fans using more elementary pieces. Namely, to each arc α we associate a shard polytope SP(α) that will ensure the presence of the shard S(α) in the normal fan of any Minkowski sum of shard polytopes containing SP(α). We introduce these shard polytopes in Section I.2.1, study their basic geometric properties in SectionI.2.2and their normal fans in SectionI.2.3, construct quotientopes as Minkowski sums of shard polytopes in SectionI.2.4, and finally observe in Section I.2.5an intriguing valuation-like Minkowski identity between shard polytopes.

I.2.1. Definition of shard polytopes. We need the following definitions. Definition 39. For an arc α:_{= (a, b, A, B), we define}

• anα-alternating matchingas a (possibly empty) sequence M = {a1, b1, . . . , ak, bk} where

a ≤ a1< b1< · · · < ak< bk ≤ b and ai∈ {a} ∪ A while bi∈ B ∪ {b} for all i ∈ [k],

• thepairsof the α-alternating matching M as the pairs (a`, b`) for ` ∈ [k],

• thecharacteristic vectorof the α-alternating matching M as χ(M ) =P

i∈[k]eai− ebi, • anα-fall(resp.α-rise) as a position j ∈ [a, b[ such that j ∈ {a} ∪ A and j + 1 ∈ B ∪ {b}

(resp. such that j ∈ {a} ∪ B and j + 1 ∈ A ∪ {b}).

Note that ∅ and {a, b} are always α-alternating matchings, and a and b − 1 are always α-falls or α-rises. Observe also that the arc α crosses the horizontal axis between the dots j and j + 1 if and only if j is an α-fall or α-rise distinct from a and b − 1. The number of α-rises plus the number of α-falls is thus the number of crossings of α with the horizontal axis plus 2.

Proposition 40. The shard polytopeSP(α) of an arc α is the polytope defined equivalently as • the convex hull of the characteristic vectors of all α-alternating matchings,

• the subset of the hyperplane H:=_{x ∈ R}n 

 P_i∈[n]xi = 0 defined by

◦ xi= 0 for any i ∈ [n] r [a, b], xa0 ≥ 0 for any a0∈ A, and x_b0 ≤ 0 for any b0∈ B, ◦ P

i≤fxi≤ 1 for any α-fall f andPi≤rxi ≥ 0 for any α-rise r.

The shard polytopes corresponding to all arcs for n = 3 and n = 4 are represented in Figures8

and 9. The α-alternating matchings are represented using solid dots • for elements in {a} ∪ A and hollow dots ◦ for elements in B ∪ {b}. This directly gives the vertex coordinates of their characteristic vectors, replacing • by 1 and ◦ by −1. For instance, the vertex labeled • · · ◦ has coordinates (1, 0, 0, −1).

Remark 41. The inequalities of Proposition40imply that 0 ≤P

i≤jxi≤ 1 for any j ∈ [n]. Indeed,

• For j ∈ [n] r [a, b[, we haveP

i≤jxi= 0 since

P

i∈[n]xi= 0 and xi= 0 for i ∈ [n] r [a, b].

• For j ∈ [a, b[, we have 0 ≤P

i≤rxi≤Pi≤jxi ≤Pi≤fxi≤ 1, where

◦ r is the last α-rise in [a, j[ while f is the first α-fall in [j, b[ if j ∈ A, ◦ r is the first α-rise in [j, b[ while f is the last α-fall in [a, j[ if j ∈ B, ◦ r is the first α-rise in [j, b[ and f is the first α-fall in [j, b[ if j = a. Considering differences of consecutive inequalities 0 ≤P

i≤jxi≤ 1, we obtain that 0 ≤ xa0 ≤ 1

for any a0 ∈ {a} ∪ A and −1 ≤ xb0 ≤ 0 for any b0 ∈ B ∪ {b}. The inequalities that we kept in Proposition40will be shown to be the facet defining inequalities of SP(α) in Proposition44(iv). Proof of Proposition40. Let P denote the V -polytope and Q denote the H-polytope defined in Proposition 40. Since the characteristic vector of any α-alternating matching clearly satisfies the given inequalities, we have P ⊆ Q. Conversely, as the equalities and inequalities defining Q have coefficients in {−1, 0, 1} and satisfy the consecutive ones property, the matrix defining Q is unimodular, so that Q has integer vertices. Consider a vertex v of Q. According to Remark41, the equalities and inequalities defining Q ensure that all the non-zero coordinates of v appear

Figure 9. Shard polytopes for all arcs with n = 4.

in between positions a and b and alternate between some 1’s at positions among {a} ∪ A and some −1’s at positions among B ∪ {b}. Therefore, v is the characteristic vector of an α-alternating

matching, and we obtain that Q ⊆ P. _

Remark 42. The vertex and facet description of Proposition 40 are somehow nicer in the ba-sis (f_i)i∈[n−1]of H defined by fi:= ei− ei+1. Namely, the shard polytope SP(α) is given by

• the 0/1-vectorsP

i∈[k]

P

j∈[ai,bi[fifor all α-alternating matchings {a1< b1< · · · < ak< bk}, • the inequalities yi = 0 for any i ∈ [n] r [a, b[, ya0 − y_a0₊₁ ≤ 0 for any a0 ∈ A while

yb0− y_b0₊₁≥ 0 for any b0∈ B, and y_f ≤ 0 for any α-fall f while y_r≥ 0 for any α-rise r. We stay in the basis (ei)i∈[n] of Rn as it will be useful to fold type A in type B in PartII.

Remark 43. To study certain properties of shard polytopes, it will sometimes be convenient to consider SP(α) for α:= (a, b, A, B) when A and B are disjoint subsets of ]a, b[ whose union does not necessarily cover ]a, b[. The vertex and facet descriptions of SP(α) are similar to Proposition40, except that

• α-alternating matchings never use elements of ]a, b[ r (A t B),

• we have the equalities xi= 0 for all i ∈ ]a, b[ r (A t B), and the α-falls (resp. α-rises) are

pairs (j, k) where j ∈ {a} ∪ A and k ∈ B ∪ {b} (resp. j ∈ {a} ∪ B and k ∈ A ∪ {b}) are consecutive in A t B, i.e. such that ]j, k[ ∩ (A t B) = ∅.

We still call SP(α) a shard polytope, even if α is not a shard, because SP(α) is affinely isomorphic to the shard polytope SP(π(α)) of the arc π(α):= (π(a), π(b), π(A), π(B)) ∈ A_{|{a,b}∪A∪B|} where π is the order preserving bijection {a, b} ∪ A ∪ B → [|{a, b} ∪ A ∪ B|].

I.2.2. Basic geometric properties of shard polytopes. The following statement, illustrated in Figure9, describes the vertices, edges and facets of the shard polytopes. Throughout the paper, we use the Kronecker notation δX= 1 if the property X holds, and δX = 0 otherwise.

Proposition 44. Let α:= (a, b, A, B) be an arc of An.

(i) The shard polytope SP(α) has dimension b − a.

(ii) The vertices of SP(α) are precisely all characteristic vectors of α-alternating matchings. Therefore, the number of vertices of SP(α) is wα(b) where vα and wα are the two

func-tions from [a, b] to N defined by the initial condifunc-tions vα(a):= wα(a):= 1 and the induction

vα(i):= vα(i−1)+δi∈{a}∪A·wα(i−1) and wα(i):= δi∈B∪{b}·vα(i−1)+wα(i−1) for i ∈ ]a, b].

(iii) The characteristic vectors of two α-alternating matchings M and M0 form an edge of SP(α) if and only if |M 4 M0| = 2. Therefore, the edge directions of SP(α) are the b−a+1

2

direc-tions ei− ej for a ≤ i < j ≤ b.

(iv) The facets of SP(α) are precisely defined by the inequalities of Proposition40. Therefore, the number of facets of SP(α) is b − a + 1 plus the number of crossings of α with the horizontal axis, and SP(α) is a simplex if and only if α does not cross the horizontal axis.

Proof. For (i), note that the shard polytope SP(α) lies in the space generated by the vectors ei−ej

for a ≤ i < j ≤ b which has dimension b − a. Conversely, SP(α) contains the b − a + 1 affinely independent vertices 0, ea− eb, ea− eb0 for b0∈ B, and e_a0− e_b for a0 ∈ A.

For (ii), the shard polytope SP(α) is the convex hull of the characteristic vectors of the α-alter-nating matchings. To see that they all define vertices, note that for any α-alterα-alter-nating match-ings M and M0, we have 2χ(M ) − 1{a}∪A+ 1B∪{b}

 χ(M0)

= |M ∩ M0| − |M0

r M |, so that χ(M ) is the only characteristic vector of an α-alternating matching that maximizes the direc-tion 2χ(M ) − 1{a}∪A+ 1B∪{b}. The induction formula for the number of α-alternating matchings

is immediate: for all i ∈ [a, b], the value vα(i) (resp. wα(i)) counts the number of odd (resp. even)

subsets of [i] of the form M ∩ [i] where M is an α-alternating matching.

For (iii), observe first that for any three α-alternating matchings M, M0 and M00, we have χ(M ) + χ(M0_{) − 1} {a}∪A+ 1B∪{b}  χ(M00)  = |M ∩ M0 ∩ M00_{| − |M}00 r (M ∪ M0)|. When |M 4 M0_{| = 2, we get χ(M ) + χ(M}0_{) − 1} {a}∪A+ 1B∪{b}  χ(M00) ≤ |M ∩ M0| with equality if and only if M00∈ {M, M0_{} (since all matchings have even cardinality).}

This shows that SP(α) has an edge joining χ(M ) to χ(M0) as soon as |M 4 M0| = 2. Conversely, assume that χ(M ) and χ(M0) form an edge in SP(α). Then the middle of χ(M ) and χ(M0) is not the middle of another pair of vertices χ(M00) and χ(M000) of SP(α). Therefore the multisets M ∪ M0and M00∪ M000 _{are distinct for any α-alternating matchings such that M, M}0_{, M}00_{, M}000 _are

all distinct. The statement then follows from the slightly more detailed combinatorial property given in Lemma49below.

For (iv), we have already seen in Proposition 40that the facets of the shard polytope SP(α) are all defined by these inequalities. Conversely, we prove that all these inequalities correspond to facets by showing that none of these inequalities is redundant. Indeed, for a0 ∈ A (resp. b0_{∈ B,}

resp. an α-fall f , resp. an α-rise r), the vector ea − ea0 (resp. e_b0 − e_b, resp. 2(e_f − e_{f +1}), resp. −er+ er+1) satisfies all these inequalities except xa0 ≥ 0 (resp. x_b0 ≤ 0, resp. P

i≤fxi≤ 1,

resp. P

i≤rxi≥ 0). The number of facets is thus b − a − 1 plus the number of α-falls plus the

number of α-rises, thus b − a + 1 plus the number of crossings of α with the horizontal axis. _ Shard polytopes also behave very nicely with respect to their faces. The following facial property is also illustrated in Figure 9. In this statement, shard polytopes are understood in the general sense of Remark43.

Proposition 45. Any face of a shard polytope is a Cartesian product of shard polytopes.

Proof. It is clearly sufficient to prove the result for facets (since an i-face is a facet of an (i+1)-face and the faces of a Cartesian product are the Cartesian products of faces of the factors). From the facet-defining inequalities of SP(α) given in Proposition40, it is immediate to observe that: